Since this is too long for a comment, I post it as an answer:

Sorry, but I do not agree with Peter Michor's answer. There are certainly better examples, but this the first I can remember: There are countable inductive limits $X=\lim\limits_{\to} X_n$ of Frechet spaces which are not Hausdorff (take a decreasing sequence of open connected sets $U_n$ in the complex plane with empty intersection and $X_n=H(U_n)$ the Frechet space of holomorphic functions on $U_n$ together with the injective restriction maps). $X$ is the a quotient of the direct sum $\bigoplus X_n$ which is certainly bornological and the kernel of the quotient map is sequentially closed
(because convergent sequences in the direct sum are located and convergent in some *finite* sum) but it is not closed because the quotient is not Hausdorff.

The situation is better for metrizable spaces (of course, this is trivial) as well as for so-called Silva spaces (also called LS or DFS-spaces, countable inductive limits of Banach spaces with compact inclusions): In these cases, sequentially closed *subspaces* are closed.

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Edit. A simpler example (but possibly less relevant for analytical applications) is the space $X=\mathbb R^I$ endowed with the product topology (point-wise convergence of functions $f:I \to \mathbb R$) if $I$ is uncountable and of moderate cardinality (e.g. $I=\mathbb R$). Then $X$ is bornological (due to the cardinality restriction) and
$L=\lbrace f\in X: \lbrace i\in I: f(i)\neq 0\rbrace \text{countable}\rbrace$ is sequentially closed and dense in $X$.